Definite Integral Properties

The initial estimate of an integral has been known as integration. It’s really the inverse differentiation process. Integrals have been used in mathematics to find many useful quantities such as areas, volumes, displacement, and so on. Integrals have been classified into two types: definite integrals and indefinite integrals. Definite integrals have been commonly used to calculate the areas of plane figures such as circles, parabolas, and ellipses.

If and only if an integral has upper and lower bounds, it is said to be a definite integral. There’s many numerous definite integral formulas and properties that are frequently used in mathematics. To calculate the value of a definite integral, find the difference between the integral’s values at the specified upper and lower limits of the independent variable, which is denoted as:

∫_a^b f(x) dx

For a function f(x) defined with reference to the x-axis, a is the lower limit and b is the upper limit. To calculate the area under a curve between two limits, divide the area into rectangles and add them together. The more rectangles there are, the more accurate the area. As a result, we divide the area into an infinite number of rectangles of the same (very small) size and add the total areas. That’s the basic theory that underpins definite integrals.

The properties of definite integrals for even and odd functions are given below. one can use these properties to solve problems involving definite integral properties.

Properties of Definite Integrals

(1) ∫_a^b f(x) dx = ∫_a^b f(t) dt

(2) ∫_a^b f(x) dx = – ∫_b^a f(x) dx … [Also, ∫_a^a f(x) dx = 0]

(3) ∫_a^b f(x) dx = ∫_a^c f(x) dx + ∫_c^b f(x) dx

(4) ∫_a^b f(x) dx = ∫_a^b f(a + b – x) dx

(5) ∫₀^a f(x) dx = ∫₀^a f(a – x) dx … [this is derived from P04]

(6) ∫₀^2a f(x) dx = ∫₀^a f(x) dx + ∫₀^a f(2a – x) dx

(7) Two parts

1. ∫₀^2af(x) dx = 2 ∫₀^a f(x) dx … if f(2a – x) = f(x).
2. ∫₀^2af(x) dx = 0 … if f(2a – x) = – f(x)

(8) Two parts

1. ∫_-a^af(x) dx = 2 ∫₀^a f(x) dx … if f(- x) = f(x) or it is an even function
2. ∫_-a^af(x) dx = 0 … if f(- x) = – f(x) or it is an odd function

FAQs

What is a Definite Integral?

Definite integrals seem to be integrals that have upper and lower bounds. Definite integrals have 2 distinct values for their upper and lower bounds. A definite integral’s final value is the value of the integral to the upper limit minus the value of the definite integral for the lower limit.

Q. What are the applications of Definite Integral?

Ans: Definite integrals have been used in many aspects of mathematics and physics; without them, no mathematical computations are possible. Definite integrals are being used to find the areas of many different plane figures such as circles, ellipses, and parabolas if the formula for their curve is provided.

• It’s being used to locate areas beneath various complex curves.
• The area between two curves can also be calculated using a definite integral.
• It is often used to compute the mass, volume, and area of various complicated three-dimensional shapes using definite integrals.